Question

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

25. \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)

Short answer

It is proved that \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\).

Step-by-step solution

Set S is affine

Theorem 2states that a set \(S\) is affineif and only if every affine combination of points of \(S\) lies in \(S\). That is, \(S\) is affine if and only if \(S = {\mathop{\rm aff}\nolimits} S\).

Show that \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)

Let \(x \in {\mathop{\rm aff}\nolimits} \left( {A \cap B} \right)\,\).

Then, \({\mathop{\rm x}\nolimits} \) can be written as the affine combination of the intersection of \(A\), and\(B\). So, \(x = {c_1}{f_1} + ... + {c_k}{f_k}\) are the elements of the intersection of \(A \cap B\), where \({c_1} + ... + {c_k} = 1\).

This means that there exist \({\mathop{\rm aff}\nolimits} A\) and \({\mathop{\rm aff}\nolimits} B\), \({f_i} \in A\,\,{\mathop{\rm and}\nolimits} \,\,\,B\,\,\,{\mathop{\rm for}\nolimits} \,\,{\mathop{\rm all}\nolimits} \,\,i\).

This implies that\({f_i} \in {\mathop{\rm aff}\nolimits} A\,\,\,{\mathop{\rm and}\nolimits} \,\,{\mathop{\rm aff}\nolimits} \,B\) for all \(i\). Therefore, \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\).

Thus, it is proved that \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\).